If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$ is closed.

Can anyone give me a trivial counter example?

  • 1
    Can you define the term 'limit point compact'? – Kavi Rama Murthy Dec 6 at 5:51
  • If $X$ is a limit point compact space then every infinite set has a limit point. E.g - $\mathbb R$ is not limit point compact space where so is any of it's closed interval.@KaviRamaMurthy – cmi Dec 6 at 5:59

Let $Y$ be $\omega_1 + 1$ and $X$ is $\omega_1$, in the order topology. Here $\omega_1$ is the first uncountable ordinal, and $Y$ is its successor (one extra point).

Another example: let $Y$ be $[0,1]^\mathbb{R}$ in the product topology, and $X$ the limit point compact subset of all elements that are $0$ except for at most countably many coordinates. ( A $\Sigma$-product, this is called). This $X$ is dense and not closed in $Y$.

  • what is first countable ordinal? Is there no simple counter example?@Henno Brandsma – cmi Dec 6 at 6:20
  • @cmi Munkres calls it $W$. Look it up – Henno Brandsma Dec 6 at 7:02

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